{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,22]],"date-time":"2025-08-22T02:10:58Z","timestamp":1755828658383,"version":"3.44.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A binary word is called $q$-decreasing, for $q&gt;0$, if inside this word each of length-maximal (in the local sense) occurrences of a factor of the form $0^a1^b$, $a&gt;0$, satisfies $q \\cdot a &gt; b$. We bijectively link $q$-decreasing words with certain prefixes of the cutting sequence of the line $y=qx$. We show that for any real positive $q$ the number of $q$-decreasing words of length $n$ grows as $C_q \\cdot \\Phi(q)^n$ for some constant $C_q$ which depends on $q$ but not on $n$. From previous works, it is already known that $\\Phi(1)$ is the golden ratio, $\\Phi(2)$ is equal to the tribonacci constant, $\\Phi(k)$ is $(k+1)$-bonacci constant. We prove that the function $\\Phi(q)$ is strictly increasing, discontinuous at every positive rational point, and exhibits a fractal structure related to the Stern-Brocot tree and Minkowski's question mark function.<\/jats:p>","DOI":"10.37236\/12705","type":"journal-article","created":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T14:05:01Z","timestamp":1755785101000},"source":"Crossref","is-referenced-by-count":0,"title":["Structure and Growth of $\\mathbb{R}$-Bonacci Words"],"prefix":"10.37236","volume":"32","author":[{"given":"Sergey","family":"Dovgal","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergey","family":"Kirgizov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,8,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/12705\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/12705\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T14:05:01Z","timestamp":1755785101000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/12705"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,22]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,7,4]]}},"URL":"https:\/\/doi.org\/10.37236\/12705","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,8,22]]},"article-number":"P3.32"}}